You can specify results that contain: The problem and options information The problem and options information, and the state of the solver when stopped this means the latest point for most solvers, the current population for Genetic Algorithms solvers, and the best point found for the Simulated Annealing solver The states of random number generators rand and randn at the start of the previous run, by checking the Use random states from previous run box for applicable solvers The options information only The results of running your problem in the Optimization app Exported results contain all optional information. For example, an exported results structure for lsqcurvefit contains the data x, resnorm, residual, exitflag, output, lambda, and jacobian. To see the value of a field in a structure or object, double-click the name in the Workspace window. Alternatively, see the values by entering exportname. For example, so see the message in an output structure, enter output. If a structure contains structures or objects, you can double-click again in the workspace browser, or enter exportname.

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Multiobjective Examples The previous examples involved problems with a single objective function. This section demonstrates solving problems with multiobjective functions using lsqnonlin , fminimax , and fgoalattain. Included is an example of how to optimize parameters in a Simulink model. This model can be found in the Optimization Toolbox optim directory.

Note that Simulink must be installed on your system to load this model. Figure Plant with Actuator Saturation The plant is an under-damped third-order model with actuator limits. The actuator limits are a saturation limit and a slew rate limit. The actuator saturation limit cuts off input values greater than 2 units or less than -2 units. The slew rate limit of the actuator is 0. You can see this response by opening the model type optsim at the command line or click the model name , and selecting Start from the Simulation menu.

The response plots to the scope. Figure Closed-Loop Response The problem is to design a feedback control loop that tracks a unit step input to the system. The closed-loop plant is entered in terms of the blocks where the plant and actuator have been placed in a hierarchical Subsystem block. A Scope block displays output trajectories during the design process. Figure Closed-Loop Model One way to solve this problem is to minimize the error between the output and the input signal.

The variables are the parameters of the PID controller. If you only need to minimize the error at one time unit, it would be a single objective function. But the goal is to minimize the error for all time steps from 0 to , thus producing a multiobjective function one function for each time step. The routine lsqnonlin is used to perform a least-squares fit on the tracking of the output.

The error signal is yout, the output computed by calling sim , minus the input signal 1. The function tracklsq must run the simulation. The simulation can be run either in the base workspace or the current workspace, i. To run the simulation in optsim, the variables Kp, Ki, Kd, a1, and a2 a1 and a2 are variables in the Plant block must all be defined.

Kp, Ki, and Kd are the variables to be optimized. You can initialize a1 and a2 before calling lsqnonlin and then pass these two variables as additional arguments. After choosing a solver using the simset function, the simulation is run using sim. The simulation is performed using a fixed-step fifth-order method to seconds.

When the simulation completes, the variables tout, xout, and yout are now in the current workspace that is, the workspace of tracklsq. The Outport block is used in the block diagram model to put yout into the current workspace at the end of the simulation. Step 1: Write an M-file tracklsq. In this case you ask for output, use the medium-scale algorithm, and give termination tolerances for the step and objective function on the order of 0. A choice must be made about the type of solver to use.

From the optimization point of view, a fixed-step solver is the best choice if that is sufficient to solve the ODE. However, in the case of a stiff system, a variable-step method might be required to solve the ODE. The numerical solution produced by a variable-step solver, however, is not a smooth function of parameters, because of step-size control mechanisms.

This lack of smoothness can prevent the optimization routine from converging. The lack of smoothness is not introduced when a fixed-step solver is used. For a further explanation, see [1].

The Nonlinear Control Design Blockset is recommended for solving multiobjective optimization problems in conjunction with variable-step solvers in Simulink. It provides a special numeric gradient computation that works with Simulink and avoids introducing a problem of lack of smoothness.

Another solution approach is to use the fminimax function. In this case, rather than minimizing the error between the output and the input signal, you minimize the maximum value of the output at any time t between 0 and Then in the function trackmmobj the objective function is simply the output yout returned by the sim command. But minimizing the maximum output at all time steps may force the output far below unity for some time steps.

To keep the output above 0. Both trackmmobj and trackmmcon use the result yout from sim , calculated from the current pid values. The nonlinear constraint function is always called immediately after the objective function in fmincon , fminimax , fgoalattain , and fseminf with the same values.

Step 1: Write an M-file trackmmobj. Step 3: Invoke constrained optimization routine. TolX and maximum constraint violation is less than options.

This solution differs from the lsqnonlin solution, because you are solving different problem formulations. The problem is to design a lowpass filter with magnitude one at all frequencies between 0 and 0. The frequency response H f for such a filter is defined by where A f is the magnitude of the frequency response. One solution is to apply a goal attainment method to the magnitude of the frequency response.

Given a function that computes the magnitude, the function fgoalattain will attempt to vary the magnitude coefficients a n until the magnitude response matches the desired response within some tolerance. The function that computes the magnitude response is given in filtmin.

This function takes a, the magnitude function coefficients, and w, the discretization of the frequency domain we are interested in. To set up a goal attainment problem, you must specify the goal and weights for the problem. For frequencies between 0 and 0. For frequencies between 0. Frequencies between 0. This information is stored in the variable goal passed to fgoalattain. The length of goal is the same as the length returned by the function filtmin. So that the goals are equally satisfied, usually weight would be set to abs goal.

Because all the goals are close in magnitude, using a weight of unity for all goals will give them equal priority. Using abs goal for the weights is more important when the magnitude of goal differs more significantly. Step 1: Write an M-file filtmin. Note that you could use the remez function in the Signal Processing Toolbox to design this filter.

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In this method, the function solves a quadratic programming QP subproblem at each iteration. The trust-region-reflective algorithm uses FiniteDifferenceType only when CheckGradients is set to true. Choose a web site to get translated content where available and see local events and offers. The outer function, nestedbowlpeakcalls fminunc and passes the objective function, nestedfun. Run the example requesting the fval output as well as the solution. Nonlinear constraints, specified as a function handle or function name.

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Multiobjective Examples The previous examples involved problems with a single objective function. This section demonstrates solving problems with multiobjective functions using lsqnonlin , fminimax , and fgoalattain. Included is an example of how to optimize parameters in a Simulink model. This model can be found in the Optimization Toolbox optim directory. Note that Simulink must be installed on your system to load this model. Figure Plant with Actuator Saturation The plant is an under-damped third-order model with actuator limits.